This is documentation for an old release of SciPy (version 1.4.1). Read this page in the documentation of the latest stable release (version 1.15.0).
Fratio (or F) Distribution¶
The distribution of \(\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right)\) if \(X_{1}\) is chi-squared with \(v_{1}\) degrees of freedom and \(X_{2}\) is chi-squared with \(v_{2}\) degrees of freedom. The support is \(x\geq0\).
\begin{eqnarray*} f\left(x;\nu_{1},\nu_{2}\right) & = & \frac{\nu_{2}^{\nu_{2}/2}\nu_{1}^{\nu_{1}/2}x^{\nu_{1}/2-1}}{\left(\nu_{2}+\nu_{1}x\right)^{\left(\nu_{1}+\nu_{2}\right)/2}B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)}\\
F\left(x;v_{1},v_{2}\right) & = & I\left(\frac{\nu_{1}x}{\nu_{2}+\nu_{1}x}; \frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)\\
G\left(q;\nu_{1},\nu_{2}\right) & = & \left(\frac{\nu_{2}} {I^{-1}\left(q; \nu_{1}/2,\nu_{2}/2\right)}-\frac{\nu_{1}}{\nu_{2}}\right)^{-1}\\
\mu & = & \frac{\nu_{2}}{\nu_{2}-2}\quad\textrm{for }\nu_{2}>2\\
\mu_{2} & = & \frac{2\nu_{2}^{2}\left(\nu_{1}+\nu_{2}-2\right)}{\nu_{1}\left(\nu_{2}-2\right)^{2}\left(\nu_{2}-4\right)}\quad\textrm{for } v_{2}>4\\
\gamma_{1} & = & \frac{2\left(2\nu_{1}+\nu_{2}-2\right)}{\nu_{2}-6}\sqrt{\frac{2\left(\nu_{2}-4\right)}{\nu_{1}\left(\nu_{1}+\nu_{2}-2\right)}}\quad\textrm{for }\nu_{2}>6\\
\gamma_{2} & = & \frac{3\left(8+\left(\nu_{2}-6\right)\gamma_{1}^{2}\right)}{2\nu-16}\quad\textrm{for }\nu_{2}>8\end{eqnarray*}
where \(I\left(x;a,b\right)=I_{x}\left(a,b\right)\) is the regularized incomplete Beta function.
Implementation: scipy.stats.f